The carpenter and Schur–Horn problems for masas in finite factors

Abstract

Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur–Horn problem, where matrix algebras and diagonals are replaced respectively, by finite factors and maximal Abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa $A$ in a finite factor $M$ with conditional expectation $\mathbb{E}_{A}$, can each $x\in A$ with $0\leq x\leq1$ be expressed as $\mathbb{E}_{A}(p)$ for a projection $p\in M$?

In this paper, we investigate these problems for various masas. We give positive solutions for the generator and radial masas in free group factors, and we also solve affirmatively a weaker form of the Schur–Horm problem for the Cartan masa in the hyperfinite factor.